^{1}

^{*}

^{1}

^{1}

^{1}

In this research work, the upward transition probabilities for the transition levels, 0
^{+} → 2
^{+}, 2
^{+} → 4
^{+}, 4
^{+} → 6
^{+} and 6
^{+} → 8
^{+} levels of even-even neutron rich
^{104-114}Ru isotopes have been calculated by using the Global Best Fit (GBF) method. In addition, the associated parameters such as, Quadrupole moment and Deformation parameter of even-even
^{104-114}Ru have been calculated. The dependency of these nuclear parameters shows the nuclear magic number tendency.

In nuclear physics, the most important part of interest is the shape of nucleus. The ground state shape of nucleus is spherical and it may deviate from this shape, which is closely related to nuclear “magic numbers”. We will consider nuclear “magic numbers” and their evolution along the nuclear chart. In stable nuclei, large gaps exist between nuclear shells when the proton or neutron number is equal to 2, 8, 20, 28, 50, 82 and 126 [^{+} states in even-even nuclei provide important information on the evolution of nuclear properties and shell model studies. The electric quadrupole reduced transition probabilities are important for the nuclear structural information. In recent years, the electric quadrupole reduced transition probabilities of even-even neutron rich nuclei ^{102-112}Pd [^{104-112}Cd [^{100-102}Ru [^{104-114}Ru and ^{106-110}Pd nuclei have been studied [^{94-110}Ru nuclei has been investigated using the Hartree-Fock-Bogoliubov (HFB) method [^{106-108}Mo and ^{108-112}Ru nuclei have been also investigated [^{104-114}Ru isotopes’ properties for the transition levels 0^{+} → 2^{+}, 2^{+} → 4^{+}, 4^{+} → 6^{+} and 6^{+} → 8^{+}. For this purpose, we have used the GBF model to investigate the basic information of the ^{104-114}Ru nuclei because this model describes three basic properties: mass and energy dependency with γ-ray transition probability; localization and emphasis for the anchor nucleus; regionalized by the magic number [

Using GBF method, we have calculated the upward transition probabilities, B(E2)↑ of neutron-rich even-even ^{104-114}Ru nuclei. Associated parameters like quadrupole moment (Q_{0}) and deformation parameter ( β 2 ) have also been estimated. The study also reveals the effects of the estimated parameters on the nuclear structure. This also reveals how the estimated parameters affect the structure of the nucleus. This method presents the near magic number N = 50 region for the ^{104}Ru nucleus. In this paper, energy and mass dependencies have been showed with the calculated B(E2)↑ values. The relationship among B(E2)↑, Q_{0} and β 2 are also given in graphical. Finally, this paper is arranged as: GBF model has been described in 2.1 Section. B(E2)↑, Q_{0} and β 2 are discussed in Section 2.2, 2.3 and 2.4 respectively.

In this section we describe the procedure used to compute the electric quadrupole reduced transition probabilities and the corresponding electric quadrupole moment and deformation parameter. The procedure summary of this theory is described in the following flowchart (

According to the Global Best Fit Method, a knowledge of the energy E (Kev) of the 2_{1}^{+} state is all that is required to make a prediction for the corresponding mean life time for the γ-ray, τ γ (in ps) and hence, the B(E2)↑ (e^{2}b^{2}) value. Within the framework of the hydrodynamic model with irrotational flow, Bohr and Mottelson [

τ γ ≈ 0.6 × 10 14 E − 4 Z − 2 A 1 / 3 (1)

For small harmonic vibrations of spherical nucleus, the τ γ value is,

τ γ ≈ 1.4 × 10 14 E − 4 Z − 2 A 1 / 3 (2)

For collective rotations of axially symmetric nuclei. The E^{−4}Z^{−2} dependence in the above expressions was adopted by Grodzins [^{1/3} with A. When the exponents of E and A were allowed to vary, we found earlier [

τ γ = 1.25 × 10 14 E − 4 Z − 2 A 0.69 (3)

Hence, τ γ and B(E2)↑ is related by the equation [

τ γ = 40.81 × 10 13 E − 5 [ B ( E 2 ) ↑ / e 2 b 2 ] − 1 (4)

When converted to B(E2)↑, this expression led to

B ( E 2 ) ↑ = 3.26 E − 1 Z 2 A − 0.69 (5)

We also showed that, the 1/E dependence is more important than the exact A dependence. If the exponent of A is fixed as −2/3 (instead of −0.69), the revised best fit to the data was found [

B ( E 2 ) ↑ = 2.6 E − 1 Z 2 A − 2 / 3 (6)

Here, B(E2)↑ is the electric quadrupole transition probability, E is the excitation energy, Z is the atomic number and A is the mass number.

The upward transition probabilities B(E2)↑ is the transition of a particle from lower energy state to higher energy state [

B ( E 2 ; l i → l f ) ↑ = 2.6 E − 1 Z 2 A − 2 / 3 (7)

Here, l i is the lower energy state and l f is the higher energy state and the subscript i and f indicate the initial and final respectively.

The nuclear electricintrinsic quadrupole moment is a parameter which describes the nuclear charge distribution. A non-zero intrinsic quadrupole moment Q_{0} indicates that the charge distribution is not spherically symmetric. By convention the value of Q_{0} is taken to be positive (Q_{0} > 0) if the ellipsoid is prolate and negative (Q_{0} < 0) if it is oblate [_{0} is related to the electric quadrupole transition probabilities B(E2)↑, calculated by the following equation [

Q 0 = [ 16 π B ( E 2 ) ↑ 5 e 2 ] 1 / 2 (8)

Here, Q_{0} measured in barn (b) unit.

Deformation Parameter is the parameter which measures the elongation of the axially symmetric shape of a deformed nucleus from its spherical shape. Deformation parameter is denoted by β 2 which is, related to B(E2)↑, calculated by the equation [

β 2 = ( 4 π / 3 Z R 0 2 ) [ B ( E 2 ) ↑ / e 2 ] 1 / 2 (9)

Here, R 0 is the average radius nuclear which can be obtained from the following equation,

R 0 2 = 0.0144 A 2 / 3 barn (b). (10)

The values of the E, R_{0}, B(E2)↑, β_{2} and Q_{0} for the even-even ^{104-114}Ru nuclei are given in ^{104-114}Ru nuclei, E has been obtained from the references [_{0}, β_{2} and R_{0}, have been obtained using the Equations (7)-(10) respectively. Using these values, Figures 2-6 have been drawn in below where their relations and behaviors have been discussed.

Electric quadrupole transition probabilities B(E2)↑ is drawn as a function of transition levels for even-even ^{104-114}Ru nuclei in ^{+} - 2^{+} transition level, transition probability is higher than that of the other transition levels for each nucleus. In this level, transition probability for the ^{104}Ru is the lowest than the other nuclei.

Deformation parameter is drawn as a function of B(E2)↑ in ^{104-114}Ru nuclei, deformation parameter change follows the almost linear relationship with respect to the transition levels.

Quadrupole moment variations with the change of B(E2)↑ are shown in

Nuclei | Transition Level, l i → l f | Energy, E in KeV [ | Average Radius, R 0 2 (b) | Upward Transition, B(E2)↑ (e^{2}b^{2}) | Deformation parameter, β 2 | Quadrupole Moment, Q_{0} (b) |
---|---|---|---|---|---|---|

^{104}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 358.02 530.46 668.12 764 | 0.31845 | 0.6357 0.4290 0.3407 0.2979 | 0.2383 0.1958 0.1744 0.1631 | 2.5279 2.0767 1.8507 1.7305 |

^{106}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 270.07 444.63 581.1 677.6 | 0.32252 | 0.8321 0.5054 0.3867 0.3316 | 0.2692 0.2098 0.1835 0.1699 | 2.8922 2.254 1.9716 1.8258 |

^{108}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 242.24 422.96 574.8 701.6 | 0.32657 | 0.9162 0.5247 0.3861 0.3163 | 0.279 0.2111 0.1811 0.1639 | 3.0349 2.2967 1.9701 1.7831 |

^{110}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 240.73 422.62 575.75 705.4 | 0.33059 | 0.9107 0.5188 0.3808 0.3108 | 0.2748 0.2074 0.1777 0.1605 | 3.0257 2.2837 1.9565 1.7676 |

^{112}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 236.66 408.24 545 649.5 | 0.33458 | 0.9153 0.5305 0.3974 0.3335 | 0.2722 0.2072 0.1793 0.1643 | 3.0334 2.3093 1.9987 1.831 |

^{114}Ru | 0^{+} - 2^{+} 2^{+} - 4^{+} 4^{+} - 6^{+} 6^{+} - 8^{+} | 265.19 443.01 590.6 709.1 | 0.33855 | 0.8073 0.4832 0.3625 0.3019 | 0.2526 0.1954 0.1693 0.1545 | 2.8488 2.204 1.9089 1.7421 |

transition probabilities increasing. In the figure, we see that quadrupole moment is lower for the isotope ^{104}Ru.

Quadrupole moments are represented with the variation of transition levels for even-even ^{104-114}Ru nuclei in

^{+} - 2^{+} the deformation of the nucleus shape will be maximum for each nucleus, gradually the deformations decrease for upper transition levels.

It is seen from the data and corresponding graphs, when the transition levels of any nuclei increased the electric quadrupole moment and reduced transition probabilities B(E2)↑ of the given nuclei are decreased. The deformation parameters also decrease with increasing transition levels. It concludes from the data and corresponding graph, the transition probabilities, quadrupole moment, and deformation parameters have comparatively lower values for the isotopes which have neutron number close to magic number 50.

The authors declare no conflicts of interest regarding the publication of this paper.

Islam, T., Amin, R., Alam, Md.A. and Islam, J. (2020) Upward Transition Probabilities B(E2)↑ Properties Study of Even-Even ^{104-114}Ru Nuclei. World Journal of Nuclear Science and Technology, 10, 129-137. https://doi.org/10.4236/wjnst.2020.103012

Upward transition probabilities B(E2)↑

Global Best Fit GBF

Quadrupole moment Q_{0}

Deformation parameter β_{2}

Interacting Boson Model-1 IBM-1

Mean life time for the γ-ray τ γ

Energy E

Proton number Z

Nuclear mass number A

Initial energy level l_{i}

Final energy level l_{f}

Electric charge e

Nuclear average radius R_{0}